Theorem iunxun 3726

Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iunxun	⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶)

Proof of Theorem iunxun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexun 3117	. . . 4 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶))
2		eliun 3652	. . . . 5 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶)
3		eliun 3652	. . . . 5 ⊢ (y ∈ ∪ x ∈ B 𝐶 ↔ ∃x ∈ B y ∈ 𝐶)
4	2, 3	orbi12i 680	. . . 4 ⊢ ((y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶) ↔ (∃x ∈ A y ∈ 𝐶 ∨ ∃x ∈ B y ∈ 𝐶))
5	1, 4	bitr4i 176	. . 3 ⊢ (∃x ∈ (A ∪ B)y ∈ 𝐶 ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶))
6		eliun 3652	. . 3 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ ∃x ∈ (A ∪ B)y ∈ 𝐶)
7		elun 3078	. . 3 ⊢ (y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶) ↔ (y ∈ ∪ x ∈ A 𝐶 ∨ y ∈ ∪ x ∈ B 𝐶))
8	5, 6, 7	3bitr4i 201	. 2 ⊢ (y ∈ ∪ x ∈ (A ∪ B)𝐶 ↔ y ∈ (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶))
9	8	eqriv 2034	1 ⊢ ∪ x ∈ (A ∪ B)𝐶 = (∪ x ∈ A 𝐶 ∪ ∪ x ∈ B 𝐶)

Syntax hints:   ∨ wo 628   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ∪ cun 2909  ∪ ciun 3648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-iun 3650

This theorem is referenced by:  iunsuc  4123  rdgisuc1  5911  oasuc  5983  omsuc  5990